We study the possible analogous of the Div-Curl Lemma in classical harmonic
analysis and partial differential equations, but from the point of view of the
multi-parameter setting. In this context we see two possible Div-Curl lemmas
that arise. Extensions to differential forms are also given.

The purposes of this paper are two fold. First, we extend the method of
non-homogeneous harmonic analysis of Nazarov, Treil and Volberg to handle
"Bergman--type" singular integral operators. The canonical example of such an
operator is the Beurling transform on the unit disc. Second, we use the methods
developed in this paper to settle the important open question about
characterizing the Carleson measures for the Besov--Sobolev space of analytic
functions $B^\sigma_2(\mathbb{B}_n)$.

Let $\Omega$ be a circular domain, that is, an open disk with finitely many
closed disjoint disks removed. Denote by $H^\infty(\Omega)$ the Banach algebra
of all bounded holomorphic functions on $\Omega$, with pointwise operations and
the supremum norm. We show that the topological stable rank of
$H^\infty(\Omega)$ is equal to 2. The proof is based on Suarez's theorem that
the topological stable rank of $H^\infty(\D)$ is equal to 2, where $\D$ is the
unit disk.